Our work is inspired by a recent idea proposed in [1]}, where a similar test is implemented for the covariate-adjusted Forward Looking Gittins Index [2]}, in a two-armed binary reward setting. In our work, we generalize the allocation probability-based test to be used with any generic randomized bandit algorithm, and we then focus on its theoretical and empirical properties when applied to TS. Notably, in addition to being a widely used and highly interpretable algorithm, we show how TS's optimal theoretical properties [3]}, [4]}, [5]} naturally translate into theoretical properties for our proposed AP-test. Such theoretical properties are not available for the FLGI in [1]}.
